Exercise 5.3

Find dy/dx in the following:

1. 2x + 3y = sin x

Solun:- Given 2x + 3y = sin x

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mn>3</mn><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mi>sin</mi><mo> </mo><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mn>2</mn><mo>+</mo><mn>3</mn><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">x</mi><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn></mrow><mn>3</mn></mfrac></math>$

2. 2x + 3y = sin y

Solun:- Given 2x + 3y = sin y

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mn>3</mn><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mi>sin</mi><mo> </mo><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mn>2</mn><mo>+</mo><mn>3</mn><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mfenced><mrow><mi>cos</mi><mo> </mo><mi mathvariant="normal">y</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>2</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mo> </mo><mi mathvariant="normal">y</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>$

3. ax + by2 = cos y

Solun:- Given ax + by2 = cos y

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi>ax</mi></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi>by</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mi>cos</mi><mo> </mo><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">a</mi><mo>+</mo><mi mathvariant="normal">b</mi><mo>×</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>×</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mo>-</mo><mi>sin</mi><mo> </mo><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mfenced><mrow><mo>-</mo><mi>sin</mi><mo> </mo><mi mathvariant="normal">y</mi><mo>-</mo><mn>2</mn><mi>yb</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal">a</mi><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mi mathvariant="normal">a</mi></mrow><mrow><mn>2</mn><mi>yb</mi><mo>+</mo><mi>sin</mi><mo> </mo><mi mathvariant="normal">y</mi></mrow></mfrac></math>$

4. xy + y2 = tan x + y

Solun:- Given xy + y2 = tan x + y

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi>xy</mi></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mi>tan</mi><mo> </mo><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi mathvariant="normal">y</mi></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">x</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi mathvariant="normal">x</mi></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>×</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi><mo>+</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">x</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>×</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi><mo>+</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi><mo>-</mo><mi mathvariant="normal">y</mi><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><msup><mi>sec</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi><mo>-</mo><mi mathvariant="normal">y</mi></mrow><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math>$

5. x2 + xy + y2 = 100

Solun:- Given x2 + xy + y2 = 100

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi>xy</mi></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mn>100</mn></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">x</mi><mfrac><mi>dx</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">x</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi><mfrac><mi>dx</mi><mi>dx</mi></mfrac><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">x</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mi mathvariant="normal">y</mi><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mi mathvariant="normal">y</mi></mrow></mfrac></math>$

6. x3 + x2y + xy2 + y3 = 81

Solun:- Given x3 + x2y + xy2 + y3 = 81

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi>xy</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">y</mi><mn>3</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mn>81</mn></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">x</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mfrac><mi>dx</mi><mi>dx</mi></mfrac><mo>+</mo><mn>3</mn><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mn>2</mn><mi>xy</mi><mo>+</mo><mn>2</mn><mi>xy</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>xy</mi><mo>+</mo><mn>3</mn><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mo>-</mo><mfenced><mrow><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>xy</mi><mo>+</mo><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>xy</mi><mo>+</mo><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>xy</mi><mo>+</mo><mn>3</mn><msup><mi mathvariant="normal">y</mi><mn>2</mn></msup></mrow></mfrac></math>$

7. sin2y + cos xy = k

Solun:- Given sin2y + cos xy = k

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi mathvariant="normal">y</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><mi>cos</mi><mo> </mo><mi>xy</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi mathvariant="normal">k</mi></mfenced></mrow><mi>dx</mi></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mn>2</mn><mi>siny</mi><mo>.</mo><mi>cosy</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mfenced><mrow><mo>-</mo><mi>sinxy</mi></mrow></mfenced><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mi>xy</mi></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>=</mo><mn>2</mn><mi>siny</mi><mo>.</mo><mi>cosy</mi><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mfenced><mrow><mo>-</mo><mi>sinxy</mi></mrow></mfenced><mfenced open="[" close="]"><mrow><mi mathvariant="normal">x</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi><mfrac><mi>dx</mi><mi>dx</mi></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mfenced><mrow><mo>-</mo><mi>sinxy</mi></mrow></mfenced><mfenced open="[" close="]"><mrow><mi mathvariant="normal">x</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mi mathvariant="normal">y</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mfenced><mrow><mo>-</mo><mi>sinxy</mi></mrow></mfenced><mo>.</mo><mi mathvariant="normal">x</mi><mo>.</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>+</mo><mfenced><mrow><mo>-</mo><mi>sinxy</mi></mrow></mfenced><mi mathvariant="normal">y</mi><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mi mathvariant="normal">x</mi><mo>.</mo><mi>sinxy</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal">y</mi><mo>.</mo><mi>sinxy</mi><mspace linebreak="newline"/><mo>⇒</mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mi>ysinxy</mi><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>-</mo><mi>xsinxy</mi></mrow></mfrac><mspace linebreak="newline"/></math>$

8. sin2x + cos2y = 1

Solun:- Given sin2x + cos2y = 1

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi>sin</mi><mo> </mo><mi mathvariant="normal">x</mi><mo>.</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">x</mi></mrow><mrow><mi>sin</mi><mo> </mo><mi mathvariant="normal">y</mi><mo>.</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">y</mi></mrow></mfrac><mspace linebreak="newline"/><mi>Multiply</mi><mo> </mo><mi>and</mi><mo> </mo><mi>divide</mi><mo> </mo><mi>by</mi><mo> </mo><mn>2</mn><mo>:</mo><mo>-</mo><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>sin</mi><mo> </mo><mi mathvariant="normal">x</mi><mo>.</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn><mi>sin</mi><mo> </mo><mi mathvariant="normal">y</mi><mo>.</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">y</mi></mrow></mfrac><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mn>2</mn><mi>sin</mi><mo> </mo><mi mathvariant="normal">x</mi><mo>.</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">x</mi><mo>=</mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">x</mi><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">y</mi></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">9</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="[" close="]"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math>$

Solun:- Given equation is:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="[" close="]"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math>$

Put x = tanθ

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="[" close="]"><mfrac><mrow><mn>2</mn><mi>tanθ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>tanθ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">θ</mi><mspace linebreak="newline"/><mspace linebreak="newline"/></math>$

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mn>2</mn><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math>$

And x = tanθ

⇒ θ = tan-1x

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo> </mo><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></math>$

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">10</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>3</mn><mi mathvariant="normal">x</mi><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup></mrow><mrow><mn>1</mn><mo>-</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo> </mo><mo>,</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><msqrt><mn>3</mn></msqrt></mfrac><mo><</mo><mi mathvariant="normal">x</mi><mo><</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math>$

Solun:- Given equation is:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>3</mn><mi mathvariant="normal">x</mi><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup></mrow><mrow><mn>1</mn><mo>-</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math>$

Put x = tanθ

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open="[" close="]"><mfrac><mrow><mn>3</mn><mi>tanθ</mi><mo>-</mo><msup><mi>tan</mi><mn>3</mn></msup><mi mathvariant="normal">θ</mi></mrow><mrow><mn>1</mn><mo>-</mo><mn>3</mn><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>3</mn><mi mathvariant="normal">θ</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>tanθ</mi><mo>-</mo><msup><mi>tan</mi><mn>3</mn></msup><mi mathvariant="normal">θ</mi></mrow><mrow><mn>1</mn><mo>-</mo><mn>3</mn><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>tan</mi><mo> </mo><mn>3</mn><mi mathvariant="normal">θ</mi></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">θ</mi><mspace linebreak="newline"/><mspace linebreak="newline"/></math>$

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mn>3</mn><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math>$

And x = tanθ

⇒ θ = tan-1x

Differentiate w.r.t. x:-

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>3</mn><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">11</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo> </mo><mo>,</mo><mn>0</mn><mo><</mo><mi mathvariant="normal">x</mi><mo><</mo><mn>1</mn></math>$

Solun:- Given equation is:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math>$

Put x = tanθ

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>cos</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">θ</mi><mspace linebreak="newline"/><mspace linebreak="newline"/></math>$

Differentiate w.r.t. x:-

And x = tanθ

⇒ θ = tan-1x

Differentiate w.r.t. x:-

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">12</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo> </mo><mo>,</mo><mn>0</mn><mo><</mo><mi mathvariant="normal">x</mi><mo><</mo><mn>1</mn></math>$

Solun:- Given equation is:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math>$

Put x = tanθ

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>cos</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>=</mo><mi>cos</mi><mo> </mo><mi mathvariant="normal">x</mi><mspace linebreak="newline"/><mspace linebreak="newline"/><mspace linebreak="newline"/></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>sin</mi><mo> </mo><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced></mrow></mfenced></mrow><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mi mathvariant="normal">θ</mi></math>$

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math>$

And x = tanθ

⇒ θ = tan-1x

Differentiate w.r.t. x:-

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">13</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo> </mo><mo>,</mo><mo>-</mo><mn>1</mn><mo><</mo><mi mathvariant="normal">x</mi><mo><</mo><mn>1</mn></math>$

Solun:- Given equation is:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math>$

Put x = tanθ

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>2</mn><mi>tanθ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>tanθ</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi></mrow></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>=</mo><mi>sin</mi><mo> </mo><mi mathvariant="normal">x</mi><mspace linebreak="newline"/><mspace linebreak="newline"/><mspace linebreak="newline"/></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>cos</mi><mo> </mo><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced></mrow></mfenced></mrow><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mi mathvariant="normal">θ</mi></math>$

Differentiate w.r.t. x:-

And x = tanθ

⇒ θ = tan-1x

Differentiate w.r.t. x:-

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">14</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mrow></mfenced><mo> </mo><mo>,</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo><</mo><mi mathvariant="normal">x</mi><mo><</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math>$

Solun:- Given equation is:-

Put x = sinθ

Differentiate w.r.t. x:-

And x = sinθ

⇒ θ = sin-1x

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo> </mo><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac></math>$

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mn>2</mn><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">15</mn><mo mathvariant="bold">.</mo><mo mathvariant="bold"> </mo><mo mathvariant="bold"> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo> </mo><mo>,</mo><mn>0</mn><mo><</mo><mi mathvariant="normal">x</mi><mo><</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math>$

Solun:- Given equation is:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math>$

Put x = cosθ

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>cos</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mspace linebreak="newline"/><mi>We</mi><mo> </mo><mi>know</mi><mo> </mo><mi>that</mi><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi><mo>=</mo><mn>2</mn><msup><mi>cos</mi><mn>2</mn></msup><mi mathvariant="normal">θ</mi><mo>-</mo><mn>1</mn><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mi>cos</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfrac></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>sec</mi><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mi mathvariant="normal">y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">θ</mi></math>$

Differentiate w.r.t. x:-

And x = cosθ

⇒ θ = cos-1x

Differentiate w.r.t. x:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mfenced><mrow><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">x</mi></mrow></mfenced></mrow><mi>dx</mi></mfrac><mo> </mo><mspace linebreak="newline"/><mo>⇒</mo><mo> </mo><mfrac><mi>dθ</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac></math>$

From Eq. 1:-

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>dy</mi><mi>dx</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn></mrow><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac></math>$

Download PDF of Exercise 5.3

NCERT Maths solutions for 12th

Important Note

Exercise 5.3

Find dy/dx in the following:

1. 2x + 3y = sin x

2. 2x + 3y = sin y

3. ax + by2 = cos y

4. xy + y2 = tan x + y

5. x2 + xy + y2 = 100

6. x3 + x2y + xy2 + y3 = 81

7. sin2y + cos xy = k

8. sin2x + cos2y = 1

$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi><mo>=</mo><msup><mi>sec</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math>$

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Important Note

Continuity & Differentiability(Ex. 5.3)

Exercise 5.3

Find dy/dx in the following:

1. 2x + 3y = sin x

2. 2x + 3y = sin y

3. ax + by2 = cos y

4. xy + y2 = tan x + y

5. x2 + xy + y2 = 100

6. x3 + x2y + xy2 + y3 = 81

7. sin2y + cos xy = k

8. sin2x + cos2y = 1

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